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Properties

Label 798.2.bc.a
Level $798$
Weight $2$
Character orbit 798.bc
Analytic conductor $6.372$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [798,2,Mod(31,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{3} + 14 q^{4} + 6 q^{5} + 4 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{3} + 14 q^{4} + 6 q^{5} + 4 q^{7} + 28 q^{9} - 2 q^{10} - 14 q^{12} - 2 q^{13} - 4 q^{14} - 6 q^{15} - 14 q^{16} + 12 q^{19} - 4 q^{21} - 12 q^{22} + 14 q^{25} - 28 q^{27} + 2 q^{28} + 18 q^{29} + 2 q^{30} - 4 q^{31} - 8 q^{34} - 10 q^{35} + 14 q^{36} - 12 q^{37} + 6 q^{38} + 2 q^{39} + 2 q^{40} + 8 q^{41} + 4 q^{42} + 10 q^{43} + 6 q^{45} + 24 q^{46} + 14 q^{48} - 4 q^{52} - 12 q^{53} - 30 q^{55} + 4 q^{56} - 12 q^{57} - 2 q^{58} + 4 q^{59} + 12 q^{62} + 4 q^{63} - 28 q^{64} + 96 q^{65} + 12 q^{66} - 6 q^{67} + 6 q^{68} - 6 q^{70} - 42 q^{71} + 12 q^{74} - 14 q^{75} + 12 q^{76} + 16 q^{77} + 12 q^{79} - 6 q^{80} + 28 q^{81} - 2 q^{84} - 8 q^{85} - 18 q^{87} - 12 q^{88} - 16 q^{89} - 2 q^{90} + 8 q^{91} + 4 q^{93} - 18 q^{94} - 82 q^{95} - 8 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i −2.48042 − 1.43207i 0.866025 + 0.500000i −0.00184546 − 2.64575i − 1.00000i 1.00000 1.43207 + 2.48042i
31.2 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i −1.27570 − 0.736523i 0.866025 + 0.500000i −1.15317 + 2.38122i − 1.00000i 1.00000 0.736523 + 1.27570i
31.3 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i −0.825412 − 0.476552i 0.866025 + 0.500000i 2.56076 − 0.665210i − 1.00000i 1.00000 0.476552 + 0.825412i
31.4 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i 0.455066 + 0.262732i 0.866025 + 0.500000i −2.53656 + 0.752229i − 1.00000i 1.00000 −0.262732 − 0.455066i
31.5 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i 1.22260 + 0.705869i 0.866025 + 0.500000i 1.61371 + 2.09665i − 1.00000i 1.00000 −0.705869 − 1.22260i
31.6 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i 1.57310 + 0.908227i 0.866025 + 0.500000i 0.235464 − 2.63525i − 1.00000i 1.00000 −0.908227 − 1.57310i
31.7 −0.866025 − 0.500000i −1.00000 0.500000 + 0.866025i 3.69678 + 2.13434i 0.866025 + 0.500000i 2.01369 + 1.71612i − 1.00000i 1.00000 −2.13434 − 3.69678i
31.8 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −3.59621 − 2.07627i −0.866025 − 0.500000i 2.46654 + 0.957163i 1.00000i 1.00000 −2.07627 − 3.59621i
31.9 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −1.33335 − 0.769808i −0.866025 − 0.500000i −2.47775 + 0.927776i 1.00000i 1.00000 −0.769808 − 1.33335i
31.10 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −1.27145 − 0.734073i −0.866025 − 0.500000i −0.0182682 − 2.64569i 1.00000i 1.00000 −0.734073 − 1.27145i
31.11 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −0.749790 − 0.432892i −0.866025 − 0.500000i −1.47106 − 2.19908i 1.00000i 1.00000 −0.432892 − 0.749790i
31.12 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i 2.08104 + 1.20149i −0.866025 − 0.500000i 0.924173 + 2.47909i 1.00000i 1.00000 1.20149 + 2.08104i
31.13 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i 2.34534 + 1.35408i −0.866025 − 0.500000i 2.44311 − 1.01548i 1.00000i 1.00000 1.35408 + 2.34534i
31.14 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i 3.15840 + 1.82350i −0.866025 − 0.500000i −2.59880 + 0.496218i 1.00000i 1.00000 1.82350 + 3.15840i
103.1 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i −2.48042 + 1.43207i 0.866025 − 0.500000i −0.00184546 + 2.64575i 1.00000i 1.00000 1.43207 − 2.48042i
103.2 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i −1.27570 + 0.736523i 0.866025 − 0.500000i −1.15317 − 2.38122i 1.00000i 1.00000 0.736523 − 1.27570i
103.3 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i −0.825412 + 0.476552i 0.866025 − 0.500000i 2.56076 + 0.665210i 1.00000i 1.00000 0.476552 − 0.825412i
103.4 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i 0.455066 − 0.262732i 0.866025 − 0.500000i −2.53656 − 0.752229i 1.00000i 1.00000 −0.262732 + 0.455066i
103.5 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i 1.22260 − 0.705869i 0.866025 − 0.500000i 1.61371 − 2.09665i 1.00000i 1.00000 −0.705869 + 1.22260i
103.6 −0.866025 + 0.500000i −1.00000 0.500000 − 0.866025i 1.57310 − 0.908227i 0.866025 − 0.500000i 0.235464 + 2.63525i 1.00000i 1.00000 −0.908227 + 1.57310i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bc.a yes 28
7.d odd 6 1 798.2.m.a ✓ 28
19.d odd 6 1 798.2.m.a ✓ 28
133.s even 6 1 inner 798.2.bc.a yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.m.a ✓ 28 7.d odd 6 1
798.2.m.a ✓ 28 19.d odd 6 1
798.2.bc.a yes 28 1.a even 1 1 trivial
798.2.bc.a yes 28 133.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} - 6 T_{5}^{27} - 24 T_{5}^{26} + 216 T_{5}^{25} + 428 T_{5}^{24} - 5034 T_{5}^{23} + \cdots + 19900521 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display